Results

The new methodology is applied to 17 years of VLBI observations in order to investigate the influ-ence of the inequality constraints on typical VLBI parameters in a long term study. In 454 out of 2333 VLBI sessions the method automatically applied inequality constraints, that means in about 20 percent of cases at least one constraint is active. Generally, the quality of the determination of baseline lengths between different VLBI telescopes is assessed in terms of baseline length repeata-bilities, which can be regarded as the standard deviation for an individual baseline after removing a linear trend from a time series of baseline lengths. For both the OLS (black) and the ICLS (red) adjustment, the baseline repeatabilities, which occur in at least 30 sessions, as well as the baseline repeatabilities for only those sessions, for which, in addition, inequality constraints are applied, are shown in Fig. 7.2(a) and Fig. 7.2(b), respectively. Further, a quadratic polynomial is fitted to the data for a better visualization. The ICLS solution (red line, Fig. 7.2(b)) is slightly more precise than the OLS solution (black line). The application of ICLS improves 9% of the baseline repeatabilities for at least 1 mm (black bars in Fig. 7.2(c)) while 1% get worse for at least 1 mm (dark gray bars) and 90% remain unchanged (light gray bars).

Concluding, the ICLS adjustment seems not to harm the estimation of the telescope positions, provided that the a priori hydrostatic component is modeled sufficiently. It is worth mentioning, that it was initially not intended to improve the determination of station coordinates, but to avoid negative tropospheric estimates and, therefore, to allow for a physically more reliable description of the atmospheric delays.

7.1. Constraining Tropospheric Delays in the VLBI Data Analysis 111

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Figure 7.2: Baseline length repeatabilities for (a) VLBI data from 1993 to 2010 and (b) those sessions, for which constraints are active, w.r.t. the classical (black) and the ICLS (red) adjustment;

and (c) difference (OLS minus ICLS) in baseline length repeatabilities(Halsig et al. 2015b).

112 7. Alternative Strategies for Modeling Atmospheric Refraction

In the following, the effect of the new methodology on zenith wet delay estimates of a single VLBI session is investigated in more detail. Since negative ZWD estimates occur most frequently in cold regions, a VLBI station in Gilmore Creek, Alaska, was selected as an example. In Fig. 7.3, the zenith wet delay estimates are illustrated for a VLBI experiment in November 2001. The ZWD parameters derived from the classical least squares adjustment are represented in black while the ICLS estimates are depicted in red. Here, the parameter referring to the second piece-wise linear segment is negative by about 3 to 4 mm and is shifted to a non-negative value in the ICLS approach.

Since continuous piece-wise linear functions are used for the parametrization of the atmospheric parameters, all zenith wet delay estimates of the same VLBI station are correlated. Although many of the ZWD estimates are also shifted, the ZWD differences between the classical and constrained adjustment are negligible, except for the parameter where the inequality constraint is active.

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Figure 7.3: Zenith wet delay parameters for the VLBI station in Gilmore Creek, Alaska in November 2001. The classical least squares estimates are represented in black and the ICLS solution is depicted in red(Halsig et al. 2016b).

Since different parameter types are possibly correlated, the influence of the single inequality con-straint (blue) applied for one zenith wet delay on other parameter groups, such as station coordinates (black), clock model correction parameters (light gray) and the zenith wet delays (dark gray) of the same station, is shown in Fig. 7.4. While the maximum difference of about 1 mm can be found in one of the ZWD estimates, the remaining part is evenly distributed between the other parameters, although the differences are on the order of tenths of a millimeter and, consequently, negligible. The effect of the same inequality constraint on the ZWD estimates of another station, as an example for Matera in Italy, is depicted in Fig. 7.4(b). The differences between both solutions are again marginal, which was confirmed by investigating other sessions, concluding that the use of the ICLS method only leads to an effect on estimates of the same station for which inequality constraints are applied.

However, this is only true if the a priori hydrostatic calibrations are modeled correctly. If the assumption of an adequate a priori model is violated, the vertical component can change noticeably

7.1. Constraining Tropospheric Delays in the VLBI Data Analysis 113

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Figure 7.4: (a) The influence of a single ZWD inequality constraint (blue) on station coordinates (black), clocks (light gray) and zenith wet delays (dark gray) of the same station (Gilmore Creek, Alaska). (b) The influence of the same inequality constraint on zenith wet delay parameters (dark gray) of another station in Matera, Italy (Halsig et al. 2016b).

in the order of several millimeters. In order to validate the influence of the meteorological data on the results, two solutions have been determined for 125 VLBI sessions in 2002, which only differ in the hydrostatic calibrations: a solution with meteorological data only derived from in-situ observations is depicted in cyan, while a solution using the combined approach described above is represented in red in Fig. 7.5. Both solutions are validated in terms of the differences in baseline length repeatabilities with respect to the classical least squares adjustment.

114 7. Alternative Strategies for Modeling Atmospheric Refraction

For about 20% of these sessions automatically inequality constraints are applied for at least one station and one zenith wet delay parameter, and approximately the same number of inequality constraints is needed for both ICLS realizations, although the constrained parameters and the or-der of magnitude in the differences to the classical least squares solution can be different. Since outliers and data gaps may occur due to sensor failures in the in-situ measurements, and inequality constraints suppress the effect of compensating erroneous hydrostatic calibrations by the zenith wet delay estimates, the baseline length repeatabilities are degraded in the case of purely in-situ measurements compared to the classical least squares solution. The situation changes when intro-ducing the strategy of combining model data and meteorological observations since the differences in baseline length repeatabilities are now negligible, although inequality constraints are introduced to allow for a more reliable estimation of tropospheric parameters.

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Figure 7.5: Baseline length repeatability differences with respect to the classical least squares so-lution. The cyan dots represent an ICLS solution, where the meteorological data used for the calculation of the hydrostatic delay only results from in-situ measurements, while red dots depict an ICLS solution using atmospheric a priori data derived from a combination of in-situ observations and a numerical weather model of the ECMWF(Halsig et al. 2016b).

In conclusion, the application of the ICLS adjustment is, in principle, possible without harming the VLBI target parameters, provided that the a priori hydrostatic component is modeled sufficiently.

However, the negative zenith wet delay estimates result not only from a priori mis-modeling, but could also be the result of several other issues, such as mis-modeling of geophysical effects as well as a certain impact due to instrumental delays or the clock parametrization. A sophisticated analysis on this topic, particularly on the stability of the hydrogen maser clocks feeding the local oscillators and other necessary electronics, is performed using close-range VLBI observations in the framework of the WHISP (Wettzell HIgh SPeed) project (see Ch. 6).